**2. Formulation of the Problem**

Consider a steady incompressible flow of Cross liquid past a vertical plate in a porous medium with slip impacts. The *x*-axis is taking along the plate and the *y*-axis perpendicular to it, as illustrated in Figure 1.

**Figure 1.** Physical diagram of the problem.

It is presumed that the free stream velocity *ue*(*x*) = *bx* and the wall temperature *Tw*(*x*) = *T*∞ + *cx* vary linearly, where *b* and *c* are two constants and *T*∞ is the temperature away from the plate. We utilize the Darcy–Forchheimer model in which the square of the velocity factor is included. In addition, the rheology equations of Cross liquid in term of viscosity are

$$\begin{cases} \tau = -\breve{p}\operatorname{I} + \operatorname{A}\_{1}\varwidetilde{\mu}(\dot{\boldsymbol{\nu}}),\\ \breve{\boldsymbol{\mu}} - \breve{\boldsymbol{\mu}}\_{\infty} = \frac{\breve{\boldsymbol{\mu}}\_{0} - \breve{\boldsymbol{\mu}}\_{\infty}}{1 + \left(\breve{\boldsymbol{\Gamma}}\dot{\boldsymbol{\gamma}}\right)^{\mathbf{N}}}, \end{cases} \tag{1}$$

Here, *n* the power-law index, Γ the time constant, A1 the first tensor of Rivlin–Ericksen and defined as A1 = (gra<sup>d</sup> V) + (gra<sup>d</sup> <sup>V</sup>)*<sup>T</sup>*, *p* the pressure, I the identity vector, .γ the rate of shear for the current model is taken as

$$
\dot{\mathbf{y}} = \left[ 4 \left( \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial \mathbf{u}}{\partial \mathbf{y}} + \frac{\partial \mathbf{v}}{\partial \mathbf{x}} \right) \right]^{1/2} \text{,} \tag{2}
$$

whereas μ0 and μ∞ represent the zero and infinite shear rates, respectively. In the present study, μ∞ is considered to be zero. Therefore, Equation (1) can be written as

$$
\widetilde{\mu} = \frac{\widetilde{\mu}\_0}{1 + \left(\widetilde{\Gamma}\dot{\boldsymbol{\gamma}}\right)^n}.\tag{3}
$$

Keeping in mind that the temperature and velocity of the two-dimensional (2D) fluid flow are considered in the forms *T* = *<sup>T</sup>*(*<sup>x</sup>*, *t*) and V = [*u*(*<sup>x</sup>*, *y*), *<sup>v</sup>*(*<sup>x</sup>*, *y*), 0], then the governing equations become

$$
\frac{\partial \upsilon}{\partial y} + \frac{\partial u}{\partial x} = 0 \tag{4}
$$

$$\frac{1}{\varepsilon^2} \left( u \frac{\partial u}{\partial \mathbf{x}} - u\_\varepsilon \frac{d u\_\varepsilon}{d \mathbf{x}} + v \frac{\partial u}{\partial y} \right) = \nu\_{eff} \frac{\partial}{\partial y} \left( \frac{\frac{\partial u}{\partial y}}{1 + \left( \Gamma \frac{\partial u}{\partial y} \right)^{1-n}} \right) - \frac{\nu(u - u\_\varepsilon)}{K\_1} - \frac{C\_F \left( u^2 - u\_\varepsilon^2 \right)}{K^{1/2}} + g \beta \tau \left( T - T\_{\rm os} \right) \tag{5}$$

$$
\alpha\_{\rm dl} \frac{\partial^2 T}{\partial y^2} - \mu \frac{\partial T}{\partial x} - v \frac{\partial T}{\partial y} = 0 \tag{6}
$$

The physical boundary conditions are

$$\begin{aligned} \mu &= L\_1 \frac{\partial \mu}{\partial y}, \; v = 0, \; T = T\_w(\mathbf{x}) + S\_1 \frac{\partial T}{\partial y} \text{ at } y = 0, \\\ u &\to u\_\epsilon(\mathbf{x}), \; T \to T\_\infty \text{ as } y \to \infty. \end{aligned} \tag{7}$$

Here, (*v*, *u*) signify, respectively, the velocity components in *x*- and *y*-directions, μ*eff* the effective (or "apparent") viscosity, <sup>ν</sup>*eff* = μ*eff* /ρ the effective kinematic viscosity, ρ the density, ε the porosity parameter, *K*1 the porous medium permeability, *k* the thermal conductivity of fluid, *CF* drag coefficient, α*m* the thermal diffusivity, *T* the temperature, *L*1 length of slip, and *S*1 proportionality constant.

Following Rosali et al. [22], we set up the similarity transformation

$$
\eta = y \sqrt{\frac{b}{a\_m}}, \quad \psi = \sqrt{b\alpha\_m} x f(\eta), \; \theta(\eta) = \frac{T - T\_{\text{cov}}}{T\_w - T\_{\text{cov}}}.\tag{8}
$$

Using the similarity transformation in the above PDEs we obtain

$$\varepsilon\_1 f^{\prime\prime\prime} \left( 1 + n \left( \mathcal{W} \varepsilon^{\prime\prime} \right)^{1-n} \right) + \begin{pmatrix} 1 + f f^{\prime\prime} - \left( f^{\prime} \right)^2 + K (1 - f^{\prime}) + \\ B \left( 1 - \left( f^{\prime} \right)^2 \right) + \lambda K \theta \end{pmatrix} \left( 1 + \left( \mathcal{W} \varepsilon^{\prime\prime} \right)^{1-n} \right)^2 = 0,\tag{9}$$

$$
\partial^{\prime\prime} + \partial^{\prime}f - \theta f^{\prime} = 0.\tag{10}
$$

The physical boundary conditions are

$$\begin{aligned} f'(0) = \gamma\_1 f''(0), \; f(0) = 0, \; \theta(0) = 1 + \gamma\_2 \theta'(0) \text{ at } \eta = 0, \\\ f'(\eta) \to 1, \; \theta(\eta) \to 0 \text{ as } \eta \to \infty. \end{aligned} \tag{11}$$

Here, the parameters are used in the above ODE's are modified porosity, dimensionless permeability, mixed convection, inertia coefficient, velocity slip, and thermal slip. These are defined as ε1 = <sup>ε</sup>2<sup>ν</sup>*eff* α*m* = ε2Pr, *K* = ε2ν *K*1*b* , λ = ε2 *g*β*T<sup>c</sup> b*2 = *Rax Pe*2*x* Pr, *B* = ε<sup>2</sup>*ueCF b* √*K* , γ1 = *L*1 + *b*α*m* , γ2 = + *b*α*m S*1. Here, *Rax* = ε2 *<sup>g</sup>*β*T*(*Tw*−*T*∞)*x*<sup>3</sup> <sup>ν</sup>*eff* α*m* is the Rayleigh number and *Pex* = *xue* α*m* is the Peclet number.

### **3. Skin Friction and Nusselt Number**

The coefficients of skin friction *Cf*and Nusselt number *Nux* are identified as

$$C\_f = \frac{2\tau\_{w}}{\rho u\_{\epsilon}^2} \text{ and } Nu\_{\mathcal{X}} = \frac{xq\_{w}}{k(T\_w - T\_{\infty})} \text{ } \tag{12}$$

where *qw* and τ*w* are identified as the heat flux and the shear stress, respectively, which are specified as

$$q\_{\rm w} = \left. -k \frac{\partial T}{\partial y} \right|\_{y=0} \text{ and } \tau\_{\rm w} = \left[ \frac{\mu \frac{\partial u}{\partial y}}{\left( 1 + \left( \Gamma \frac{\partial u}{\partial y} \right)^{1-n} \right)} \right]\_{y=0}. \tag{13}$$

*Mathematics* **2020**, *8*, 31

> Utilizing Equation (8), we have

$$\frac{1}{2}\sqrt{\frac{\text{Re}\_{\text{x}}}{\text{Pr}}}\mathbb{C}\_{f} = \frac{f^{\prime\prime}(0)}{\left(1 + \left(N\epsilon f^{\prime\prime}(0)\right)^{1-n}\right)} \text{ and } -\theta^{\prime}(0) = \frac{Nu\_{\text{x}}}{\sqrt{\text{Pr}\_{\text{x}}}}.\tag{14}$$
